calculus

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i understand optimization but im stuck on this problem.. mrs.day is building a new deck. she has 580 square feet to enclose. if one side is bounded by a wall, find the minimum cost to build the deck if one pair of opposite sides cost \$12 per foot and remaining sides cost \$19 per foot. what are the demisions of the deck?

• calculus -

Is she building a deck or a railing around the deck.
From the wording I will assume a railing.
The two sides of the railing, length a, cost 12/ft each.
The front of the railing, length b, costs 19.ft

a*b = 580

2*12 a + 19 b = c, the cost

dc/da = 0 for min = 24 da/da +19 db/da
but
b = 580/a
db/da = -580/a^2
so
0 = 24 +19 (-580)/a^2
24 a^2 = 19*590
a^2 = 459
a = 21.4
then b = 27
Now it might be that the front cost 12 and the sides cost 19
in that case
2*19 a + 12 b = c
dc/da = 0 = 38 da/da + 12 db/da
db/da is still -580/a^2
38 = 12(580)/a^2
a^2 = 183
a = 13.5
then b = 42.8

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